Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces

Authors: J. W. Barrett; R. Nürnberg

Source: IMA Journal of Numerical Analysis, Volume 24, Number 2, April 2004 , pp. 323-363(41)

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Abstract:

We consider a fully practical finite-element approximation of the following system of nonlinear degenerate parabolic equations:

(

u)/(

t) + ½

. (u²

[

(v)]) - (1)/(3) nabla

.(u³

w)= 0,

w = - c

u -

u^-+ a u^-3 ,

(

v)/(

t) +

. (u v

[

(v)]) -

v - ½

.(u² v

w) = 0.

The above models a surfactant-driven thin-film flow in the presence of both attractive, a>0, and repulsive, dgr

>0 with

>3, van der Waals forces; where u is the height of the film, v is the concentration of the insoluble surfactant monolayer and sgr

(v):=1-v is the typical surface tension. Here rgr

0 and c>0 are the inverses of the surface Peclet number and the modified capillary number. In addition to showing stability bounds for our approximation, we prove convergence, and hence existence of a solution to this nonlinear degenerate parabolic system, (i) in one space dimension when rgr

>0; and, moreover, (ii) in two space dimensions if in addition ngr

7. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.

Keywords: thin film flow; surfactant; fourth order degenerate parabolic system; finite elements; convergence analysis

Document Type: Research article

Affiliations: 1: Department of Mathematics, Imperial College, London SW7 2AZ, UK

Publication date: 2004-04-01

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The IMA Journal of Numerical Analysis (IMAJNA) publishes original contributions to all fields of numerical analysis; articles will be accepted which treat the theory, development or use of practical algorithms and interactions between these aspects. Occasional survey articles are also published.